Coupling of multiscale lattice Boltzmann discrete-element method for reactive particle fluid flows

Reactive particulate systems are of prime importance in varieties of practical applications in process engineering. As an example this study considers extraction of phosphorous from waste water by calcium silicate hydrate particles in the P-RoC process. For such systems modeling has a large potential to help to optimize process conditions, e.g., particle-size distributions or volume flows. The goal of this study is to present a new generic modeling framework to capture relevant aspects of reactive particle fluid flows using combined lattice Boltzmann method and discrete-element method. The model developed is Euler-Lagrange scheme which consist of three-components viz., a fluid phase, a dissolved reactive substance, and suspended particles. The fluid flow and reactive mass transport are described in a continuum framework using volume-averaged Navier-Stokes and volume-averaged advection-diffusion-reaction equations, respectively, and solved using lattice Boltzmann methods. The volume-averaging procedure ensures correctness in coupling between fluid flow, reactive mass transport, and particle motion. The developed model is validated through series of well-defined benchmarks. The benchmarks include the validation of the model with experimental data for the settling of a single particle in a cavity filled with water. The benchmark to validate the multi-scale reactive transport involves comparing the results with a resolved numerical simulation. These benchmarks also prove that the proposed model is grid convergent which has previously not been established for such coupled models. Finally, we demonstrate the applicability of our model by simulating a suspension of multiple particles in fluid with a dissolved reactive substance. Comparison of this coupled model is made with a one-way coupled simulation where the influence of particles on the fluid flow and the reactive solution transport is not considered. This elucidates the need for the two-way coupled model.

Figure 1

Scheme of the fully coupled system of fluid, reactive substance, and particles.

Figure 2

Physical setup of the single-particle settling experiment.

Figure 3

The averaged velocity $\langle u_2\rangle$ of the fluid in $y$ direction is accelerated by the settling of the medium-sized particle. Four different times are depicted in the centered $x\text{−}y$ plane ($\mathrm{\Delta }t=5.62\times {10}^{-4}$ s, $h=200$).

Figure 4

Depending on the size of the violet colored particle small vortices in the fluid arise next to it [(a) $r_l=2.5\times {10}^{-4}$ m, (b) $r_l=7.5\times {10}^{-4}$ m, (c) $r_l=1.0\times {10}^{-3}$ m, $\mathrm{\Delta }t=5.62\times {10}^{-4}$ s, $h=200$]. The vectors located at every Euler grid point depict the direction and are scaled by a multiple of the magnitude of the fluid velocity.

Figure 5

For the medium-sized particle, the settling velocity over time differs little for simulations of different $\mathrm{\Delta }t$. A convergence of the velocity towards the results of the simulation with the finest $\mathrm{\Delta }t$ is obvious by a more detailed look.

Figure 6

(a) The relative error ${\text{err}}_p\left(h\right)$ is plotted over the time increment. It is smaller than 0.3%. The slope of the fit to ${\text{err}}_p$, the particle EOC, is $\sim 1.5$. (b) The simulation converges for decreasing $\mathrm{\Delta }x$. The slope of the fit to ${\text{err}}_f$, the fluid EOC, is $\sim 2$.

Figure 7

The fast increase of the particle settling velocity generates kind of a momentum perturbation. This results in a noise throughout the whole domain with values of about $2.5\times {10}^{-5}$ m ${\mathrm{s}}^{-1}$ for $h=200$ for the medium-sized particle at $t=1$ s.

Figure 8

The validation test case consists of a rectangular domain in 2D and contains 50 spherical particles where the reaction takes place. The dashed lines depict the computational grid for the coarsest resolution ($\mathrm{\Delta }x=5\times {10}^{-4}\mathrm{m}$).

Figure 9

Comparison of the resolved and the unresolved particle simulations for about $2\times {10}^{-4}\mathrm{s}$.

Figure 10

Comparison of the resolved (a) and the unresolved (b) particle simulations for about $2\times {10}^{-4}\mathrm{s}$ ($\mathrm{\Delta }x=5\times {10}^{-4}\mathrm{m}$) in a filled contour plot. In panel (b), the position of the particles is indicated by a white dotted line, the underlying grid is given by the black dotted lines.

Figure 11

The relative error ${\text{err}}_c$ of the concentration at the outlet at time $t=2\times {10}^{-4}\mathrm{s}$ is smaller than $\sim 0.1\%$. The slope of the fit to ${\text{err}}_c$ is about 2.

Figure 12

Discretization of the simulation domain and depiction of 2000 particles. Some particles protrude from the box because of the periodicity of the boundaries in $x$, $y$, and $z$ direction for fluid and particles (left). In case of the concentration the boundaries are periodic in $y$ and $z$ direction. Crop of one Euler grid cell that contains three Lagrange particles (right).

Figure 13

Three dimensional depiction of the concentration $c$ after $t=2.4\mathrm{s}$ for 2000 particles (a) and magnitude of the associated velocity $\parallel \vec{u}\parallel$ (b).

Figure 14

Temporal progress of the mean concentration $c_{\mathrm{out}}$ at the outlet of the channel segment. For both the one- and the two-way coupling, the concentration decreases because of the reaction that takes place at the surface of the 1000, 2000, and 3000 particles.